Book contents
- Frontmatter
- Contents
- Preface
- 1 Linear algebra
- 2 Multilinear algebra
- 3 Differentiation on manifolds
- 4 Homotopy and de Rham cohomology
- 5 Elementary homology theory
- 6 Integration on manifolds
- 7 Vector bundles
- 8 Geometric manifolds
- 9 The degree of a smooth map
- Appendix A Mathematical background
- Appendix B The spectral theorem
- Appendix C Orientations and top-dimensional forms
- Appendix D Riemann normal coordinates
- Appendix E Holonomy of an infinitesimal loop
- Appendix F Frobenius' theorem
- Appendix G The topology of electrical circuits
- Appendix H Intrinsic and extrinsic curvature
- References
- Index
Appendix A - Mathematical background
Published online by Cambridge University Press: 05 June 2014
- Frontmatter
- Contents
- Preface
- 1 Linear algebra
- 2 Multilinear algebra
- 3 Differentiation on manifolds
- 4 Homotopy and de Rham cohomology
- 5 Elementary homology theory
- 6 Integration on manifolds
- 7 Vector bundles
- 8 Geometric manifolds
- 9 The degree of a smooth map
- Appendix A Mathematical background
- Appendix B The spectral theorem
- Appendix C Orientations and top-dimensional forms
- Appendix D Riemann normal coordinates
- Appendix E Holonomy of an infinitesimal loop
- Appendix F Frobenius' theorem
- Appendix G The topology of electrical circuits
- Appendix H Intrinsic and extrinsic curvature
- References
- Index
Summary
A.1 Sets and maps
A setX is a collection of objects, which we call elements. We write x ∈ X if x is an element of X. We also write X = {x, y,…} to denote the elements of X. The empty set ∅ = {} is the unique set containing no elements. A set U is a subset of X, written U ⊆ X, if x ∈ U implies x ∈ X. A set U is a proper subset of X, written U ⊂ X, if U ⊆ X and U ≠ X. The unionX ∪ Y of two sets X and Y is the set of all elements in X or in Y (or in both), whereas the intersectionX ∩ Y is the set of all elements that are in both X and Y. If X ∩ Y = ∅ then XmeetsY (and Y meets X). The (set-theoretic) difference of two sets X and Y is the set X\Y = X − Y = {x ∈ X : x ∉ Y}. (This definition does not require that Y be contained in X.) The complement of U ⊆ X is Ū ≔ X − U.
- Type
- Chapter
- Information
- Manifolds, Tensors, and FormsAn Introduction for Mathematicians and Physicists, pp. 263 - 270Publisher: Cambridge University PressPrint publication year: 2013